Theorem th3q 6234

Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
th3q.1	|- .~ e. _V
th3q.2	|- .~ Er ( S X. S )
th3q.4	|- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
th3q.5	|- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }

Assertion

Ref	Expression
th3q	|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )

Distinct variable groups:   x, y, z, w, v, u, t, s, f, g, h, .~   x, S, y, z, w, v, u, t, s, f, g, h   x, A, y, z, w, v, u, t, s, f   x, B, y, z, w, v, u, t, s, f   x, C, y, z, w, v, u, t   x, D, y, z, w, v, u, t   x, .+ , y, z, w, v, u, t, s, f, g, h

Allowed substitution hints:   A( g, h)   B( g, h)   C( f, g, h, s)   D( f, g, h, s)   G( x, y, z, w, v, u, t, f, g, h, s)

Proof of Theorem th3q

Step	Hyp	Ref	Expression
1		opelxpi 4394	. . . 4 |- ( ( A e. S /\ B e. S ) -> <. A , B >. e. ( S X. S ) )
2		th3q.1	. . . . 5 |- .~ e. _V
3	2	ecelqsi 6183	. . . 4 |- ( <. A , B >. e. ( S X. S ) -> [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) )
4	1, 3	syl 14	. . 3 |- ( ( A e. S /\ B e. S ) -> [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) )
5		opelxpi 4394	. . . 4 |- ( ( C e. S /\ D e. S ) -> <. C , D >. e. ( S X. S ) )
6	2	ecelqsi 6183	. . . 4 |- ( <. C , D >. e. ( S X. S ) -> [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) )
7	5, 6	syl 14	. . 3 |- ( ( C e. S /\ D e. S ) -> [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) )
8	4, 7	anim12i 331	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) ) )
9		eqid 2081	. . . 4 |- [ <. A , B >. ] .~ = [ <. A , B >. ] .~
10		eqid 2081	. . . 4 |- [ <. C , D >. ] .~ = [ <. C , D >. ] .~
11	9, 10	pm3.2i 266	. . 3 |- ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ )
12		eqid 2081	. . 3 |- [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~
13		opeq12 3572	. . . . . 6 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14		eceq1 6164	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> [ <. w , v >. ] .~ = [ <. A , B >. ] .~ )
15	14	eqeq2d 2092	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ <-> [ <. A , B >. ] .~ = [ <. A , B >. ] .~ ) )
16	15	anbi1d 452	. . . . . . 7 |- ( <. w , v >. = <. A , B >. -> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) ) )
17		oveq1 5539	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> ( <. w , v >. .+ <. C , D >. ) = ( <. A , B >. .+ <. C , D >. ) )
18	17	eceq1d 6165	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> [ ( <. w , v >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )
19	18	eqeq2d 2092	. . . . . . 7 |- ( <. w , v >. = <. A , B >. -> ( [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) )
20	16, 19	anbi12d 456	. . . . . 6 |- ( <. w , v >. = <. A , B >. -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) ) )
21	13, 20	syl 14	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) ) )
22	21	spc2egv 2687	. . . 4 |- ( ( A e. S /\ B e. S ) -> ( ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> E. w E. v ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) ) )
23		opeq12 3572	. . . . . . 7 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
24		eceq1 6164	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> [ <. u , t >. ] .~ = [ <. C , D >. ] .~ )
25	24	eqeq2d 2092	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> ( [ <. C , D >. ] .~ = [ <. u , t >. ] .~ <-> [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) )
26	25	anbi2d 451	. . . . . . . 8 |- ( <. u , t >. = <. C , D >. -> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) ) )
27		oveq2 5540	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> ( <. w , v >. .+ <. u , t >. ) = ( <. w , v >. .+ <. C , D >. ) )
28	27	eceq1d 6165	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> [ ( <. w , v >. .+ <. u , t >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ )
29	28	eqeq2d 2092	. . . . . . . 8 |- ( <. u , t >. = <. C , D >. -> ( [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) )
30	26, 29	anbi12d 456	. . . . . . 7 |- ( <. u , t >. = <. C , D >. -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) ) )
31	23, 30	syl 14	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) ) )
32	31	spc2egv 2687	. . . . 5 |- ( ( C e. S /\ D e. S ) -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) -> E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
33	32	2eximdv 1803	. . . 4 |- ( ( C e. S /\ D e. S ) -> ( E. w E. v ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
34	22, 33	sylan9 401	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
35	11, 12, 34	mp2ani 422	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
36		ecexg 6133	. . . 4 |- ( .~ e. _V -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V )
37	2, 36	ax-mp 7	. . 3 |- [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V
38		eqeq1 2087	. . . . . . . 8 |- ( x = [ <. A , B >. ] .~ -> ( x = [ <. w , v >. ] .~ <-> [ <. A , B >. ] .~ = [ <. w , v >. ] .~ ) )
39		eqeq1 2087	. . . . . . . 8 |- ( y = [ <. C , D >. ] .~ -> ( y = [ <. u , t >. ] .~ <-> [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) )
40	38, 39	bi2anan9 570	. . . . . . 7 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ ) -> ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) ) )
41		eqeq1 2087	. . . . . . 7 |- ( z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ -> ( z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
42	40, 41	bi2anan9 570	. . . . . 6 |- ( ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ ) /\ z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> ( ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
43	42	3impa 1133	. . . . 5 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ /\ z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> ( ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
44	43	4exbidv 1791	. . . 4 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ /\ z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> ( E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
45		th3q.2	. . . . 5 |- .~ Er ( S X. S )
46		th3q.4	. . . . 5 |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
47	2, 45, 46	th3qlem2 6232	. . . 4 |- ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
48		th3q.5	. . . 4 |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }
49	44, 47, 48	ovig 5642	. . 3 |- ( ( [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) )
50	37, 49	mp3an3 1257	. 2 |- ( ( [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) )
51	8, 35, 50	sylc 61	1 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   <.cop 3401   class class class wbr 3785   X. cxp 4361  (class class class)co 5532   {coprab 5533   Er wer 6126   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by:  oviec  6235